Capacity of random channels with large alphabets

Figure 1.  For different alphabet sizes $n$ we plot the capacity of five random channels, constructed as explained in Example 3.1. The method introduced in [20] is used to determine upper and lower bounds for the capacity for finite alphabet sizes $n$. The asymptotic capacity (for $n\to \infty$) is depicted by the dashed line

Figure 2.  For different alphabet sizes $n$, we plot in (a) the empirical mean of the maximum expected information gain (blue line) $\tfrac{1}{N} \sum_{i=1}^N \sup_{\lambda\in\Lambda} I(p, \mathsf{W}_i^{(\lambda, V, n)})$, where $(\mathsf{W}_i^{(\lambda, V, n)})_{i=1}^N$ are independent channels and $N=1000$. The red line represents the empirical mean of the suboptimal expected information gain, that is given by $\tfrac{1}{N} \sum_{i=1}^N I(p, \mathsf{W}_i^{(\hat{\lambda}, V, n)})$, where $\hat{\lambda}$ are the optimal parameters for the asymptotic capacity, derived in Proposition 4.3. (b) depicts the empirical variance of the maximum expected information gain (blue line) as well as the empirical variance of the suboptimal expected information gain (red line)